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2009 | OriginalPaper | Buchkapitel
Some basic inequalities
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As we all know, one of the important properties of real numbers is comparability. We can compare two distinct real numbers and say that one is smaller or larger than the other. There is an inherent ordering < on the real number system ℝ which helps us to compare two real numbers. The basic properties of this ordering on ℝ are: (i)Given any two real numbers a and b, one and only one of the following three relations is true: <math display='block'> <mrow> <mi>a</mi><mo><</mo><mi>b</mi><mtext> or </mtext><mi>a</mi><mo>=</mo><mi>b</mi><mtext> or </mtext><mi>a</mi><mo>></mo><mi>b</mi><mo>;</mo> </mrow> </math>$$a < b{\text{ or }}a = b{\text{ or }}a > b;$$ (law of trichotomy.)(ii)a > 0 and b > 0 imply a + b > 0;(iii)a > 0 and b > 0 imply ab > 0.